View source for Sylow Theory



Let G be a finite group and p a prime. Write the order of <math> G</math> as
<math> |G| = p^em </math> where <math>p \not| m</math>. A Sylow p-subgroup of <math> G</math> is a subgroup <math> H \leq G </math>such that <math> |H| = p^e</math>. That is, a Sylow p-subgroup of <math>G </math> is a subgroup whose order is the highest conceivable power of <math>p </math> according to Lagrange’s Theorem.
We set <math> \text{Syl}_p(G)</math> to be the collection of all Sylow p-subgroups of <math>G </math> and <math>n_p = | \text{Syl}_p(G)| </math> to be the number of Sylow p-subgroups.